Problem: If the graph of the line $y = ax + b$ passes through the points $(4,5)$ and $(8,17)$, what is $a - b$?
The slope of a line through two points, $(x_1,y_1)$ and $(x_2,y_2)$, is \[\frac{y_2 - y_1}{x_2 - x_1}.\]Let $(x_1,y_1) = (4,5)$ and $(x_2,y_2) = (8,17)$. Then the slope of the line through the two points is \[\frac{y_2 - y_1}{x_2 - x_1} = \frac{17 - 5}{8 - 4} = \frac{12}{4} = 3.\]Thus, $a = 3$.

$b$ satisfies $y = 3x + b$ for all points on its graph. Since $(4,5)$ lies on the the graph of $y = 3x + 5$, we can substitute $x = 4$ and $y = 5$ to solve for $b$. $5 = 3(4) + b$ and subtracting 12 from both sides yields $b = -7$. Therefore, $a - b = 3 - (-7) = \boxed{10}$.